Prove that `[lim_(xto0) (tan^(-1)x)/(x)]=0,` where `[.]` represents the greatest integer function.

Explanation

To prove that \(\lim_{x \to 0} \frac{\tan^{-1} x}{x} = 0\) and that the greatest integer function \(\lfloor \lim_{x \to 0} \frac{\tan^{-1} x}{x} \rfloor = 0\), we will analyze the limit from both sides (right-hand limit and left-hand limit).

### Step 1: Analyze the right-hand limit as \(x\) approaches 0 from the positive side
We start by considering the limit:
\[ \lim_{x \to 0^+} \frac{\tan^{-1} x}{x} \]
We know that \(\tan^{-1} x < x\) for \(x > 0\) and close to 0. Therefore, we can write:
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